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voltage relationships in the stationary a-b-c reference frame are e_a = Ecos(wt) = -\cfrac{di_a}{dt} + v_{an}

e_b = Ecos(wt-\cfrac{2}{3}\pi) = -\cfrac{di_a}{dt} + v_{bn}

e_c = Ecos(wt+\cfrac{2}{3}\pi) = -\cfrac{di_a}{dt} + v_{cn}

In synchronous d-q reference frame as

e_d = E = -L \cfrac{di_d}{dt} + \omega Li_q + v_d e_q = 0 \ -L \cfrac{di_q}{dt} - \omega Li_d + v_q

For fast current dynamics, current controller are designed as

v_d* = E - \omega Li_q + \varDelta v_d

v_q* = \omega Li_d + \varDelta v_q

where \varDelta v_d , \varDelta v_q are

\varDelta v_d = k_{pd}(i_d^* - i_d) + k_{id} \int (i_d^* - i_d) dt

\varDelta v_q = k_{pq}(i_q^* - i_q) + k_{iq} \int (i_q^* - i_q) dt


m_a = K_{i}\int(i_d^* - i_d)dt

U_{d} = K_{pd} \cdot (i_d^* - i_d) + \left( m_a \cdot V_d\right)

U_{q} = K_{pq} \cdot (i_q^* - i_q) + \left( m_a \cdot V_q\right)